Tutorial: Floating-Point Binary

The two most common floating-point binary storage
formats used by Intel processors were created for Intel and later standardized
by the IEEE organization:

IEEE Short Real: 32 bits

1 bit for the sign, 8 bits for the exponent,
and 23 bits for the mantissa. Also called single precision.

IEEE Long Real: 64 bits

1 bit for the sign, 11 bits for the exponent,
and 52 bits for the mantissa. Also called double precision.

Both formats use essentially the same method for storing floating-point binary numbers, so
we will use the Short Real as an example in this tutorial. The bits in an IEEE
Short Real are arranged as follows, with the most significant bit (MSB) on the left:

Fig. 1

The Sign

The sign of a binary floating-point number is represented by a
single bit. A 1 bit indicates a negative number, and a 0 bit indicates a positive number.

The Mantissa

It is useful to consider the way decimal floating-point numbers
represent their mantissa. Using -3.154 x 105 as an example, the sign
is negative, the mantissa is 3.154, and the exponent is
5. The fractional portion of the mantissa is the sum of each digit multiplied by a power
of 10:

.154 = 1/10 + 5/100 + 4/1000

A binary floating-point number is similar. For example, in the
number +11.1011 x 23, the sign is positive, the mantissa is
11.1011, and the exponent is 3. The fractional portion of the mantissa is the sum of
successive powers of 2. In our example, it is expressed as:

.1011 = 1/2 + 0/4 + 1/8 + 1/16

Or, you can calculate this value as 1011 divided by 24.
In decimal terms, this is eleven divided by sixteen, or 0.6875. Combined with the
left-hand side of 11.1011, the decimal value of the number is 3.6875. Here are additional
examples:

Binary Floating-Point

Base 10 Fraction

Base 10 Decimal

11.11

3 3/4

3.75

0.00000000000000000000001

1/8388608

0.00000011920928955078125

The last entry in this table shows the smallest fraction that can
be stored in a 23-bit mantissa. The following table shows a few simple examples of binary
floating-point numbers alongside their equivalent decimal fractions and decimal values:

Binary

Decimal Fraction

Decimal Value

.1

1/2

.5

.01

1/4

.25

.001

1/8

.125

.0001

1/16

.0625

.00001

1/32

.03125

The Exponent

IEEE Short Real exponents are stored as 8-bit unsigned
integers with a bias of 127. Let's use the number 1.101 x 25 as an
example. The exponent (5) is added to 127 and the sum (132) is binary 10000100.
Here are some examples of exponents, first shown in decimal, then adjusted,
and finally in unsigned binary:

Exponent
(E)

Adjusted
(E + 127)

Binary

+5

132

10000100

0

127

01111111

-10

117

01110101

+128

255

11111111

-127

0

00000000

-1

126

01111110

The binary exponent is unsigned, and therefore cannot
be negative. The largest possible exponent is 128-- when added to 127, it produces
255, the largest unsigned value represented by 8 bits. The approximate range
is from 1.0 x 2-127 to 1.0 x 2+128.

Normalizing the Mantissa

Before a floating-point binary number can be stored correctly, its
mantissa must be normalized. The process is basically the same as when normalizing a
floating-point decimal number. For example, decimal 1234.567 is normalized as 1.234567 x
103 by moving the decimal point so that only one digit appears before the
decimal. The exponent expresses the number of positions the decimal point was moved left
(positive exponent) or moved right (negative exponent).

Similarly, the floating-point binary value 1101.101 is normalized
as 1.101101 x 23 by moving the decimal point 3 positions to the left, and
multiplying by 23. Here are some examples of normalizations:

Binary Value

Normalized As

Exponent

1101.101

1.101101

3

.00101

1.01

-3

1.0001

1.0001

0

10000011.0

1.0000011

7

You may have noticed that in a normalized mantissa, the
digit 1 always appears to the left of the decimal point. In fact, the leading
1 is omitted from the mantissa in the IEEE storage format because it is redundant.

Creating the IEEE Bit Representation

We can now combine the sign, exponent, and normalized
mantissa into the binary IEEE short real representation. Using Figure 1 as a
reference, the value 1.101 x 20 is stored as sign = 0 (positive),
mantissa = 101, and exponent = 01111111 (the exponent value is added to 127).
The "1" to the left of the decimal point is dropped from the mantissa.
Here are more examples:

Binary Value

Biased Exponent

Sign, Exponent, Mantissa

-1.11

127

1 01111111 11000000000000000000000

+1101.101

130

0 10000010 10110100000000000000000

-.00101

124

1 01111100 01000000000000000000000

+100111.0

132

0 10000100 00111000000000000000000

+.0000001101011

120

0 01111000 10101100000000000000000

Converting Decimal Fractions to Binary Reals

If a decimal fraction can be easily represented as a sum of fractions in the
form (1/2 + 1/4 + 1/8 + ... ), it is fairly easy to discover the corresponding
binary real. Here are a few simple examples

Decimal Fraction

Factored As...

Binary
Real

1/2

1/2

.1

1/4

1/4

.01

3/4

1/2 + 1/4

.11

1/8

1/8

.001

7/8

1/2 + 1/4 + 1/8

.111

3/8

1/4 + 1/8

.011

1/16

1/16

.0001

3/16

1/8 + 1/16

.0011

5/16

1/4 + 1/16

.0101

Of course, the real world is never so simple. A fraction such as 1/5 (0.2)
must be represented by a sum of fractions whose denominators are powers of 2.
Here is the output from a program that subtracts each succesive fraction from
0.2 and shows each remainder. In fact, an exact value is not found after creating
the 23 mantissa bits. The result, however, is accurate to 7 digits. The blank
lines are for fractions that were too large to be subtracted from the remaining
value of the number. Bit 1, for example, was equal to .5 (1/2), which could
not be subtracted from 0.2.